#8119894

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10)(11,12)(13,20)(14,22)(15,16)(17,23)(18,24)(19,21)(26,30)(27,33)(32,37)(34,39);(2,5)(6,7)(9,11,15,13)(10,12,16,20)(14,18,23,21)(17,19,22,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,2)(3,6)(4,5)(7,8)(10,16)(11,13)(17,22)(18,21)(28,35)(32,37)(34,39)(38,40);(1,3,4,8)(2,6,5,7)(9,12)(10,11)(13,16)(14,19)(15,20)(17,18)(21,22)(23,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(9,13,15,11)(10,12,16,20)(14,21,23,18)(17,19,22,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(9,14,15,23)(10,17,16,22)(11,18,13,21)(12,19,20,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40)

Polycyclic presentation:

2^9|b^2=h,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8027090);